Steps on the way to Lightcone cosmological calculator

[marcus]

Here's a sample cosmic history using the new Planck numbers. In Jorrie's cosmic tabulator I put
the Hubble expansion times 14.56 and 17.6, and the crossover S=3400, as per the Planck report.
Then to specify the dimensions of the table, I said
upper=45
lower=0.04
steps=20
and checked the "S = exactly 1" box.
This means the table goes back in past well before stars existed, when distances were 1/45 what they are today, and it goes out into future when distances are 25 times what they are today.
I also set all the columns to have 3-place precision. The often useful 6-place precision was not needed in this case.

{\begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.56&17.6&3400&67.17&0.684&0.316\\ \hline\end{array}} {\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&T_{Hub}(Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline45.000&0.022&0.056&0.085&39.362&0.875&1.247&0.153\\ \hline37.201&0.027&0.075&0.114&38.595&1.037&1.488&0.206\\ \hline30.753&0.033&0.100&0.151&37.750&1.228&1.772&0.277\\ \hline25.423&0.039&0.133&0.201&36.821&1.448&2.107&0.372\\ \hline21.017&0.048&0.178&0.268&35.798&1.703&2.500&0.498\\ \hline17.374&0.058&0.237&0.357&34.672&1.996&2.959&0.667\\ \hline14.363&0.070&0.316&0.475&33.434&2.328&3.494&0.893\\ \hline11.874&0.084&0.420&0.632&32.071&2.701&4.111&1.196\\ \hline9.816&0.102&0.559&0.841&30.573&3.115&4.821&1.599\\ \hline8.115&0.123&0.745&1.118&28.926&3.565&5.628&2.137\\ \hline6.708&0.149&0.991&1.485&27.115&4.042&6.538&2.855\\ \hline5.546&0.180&1.317&1.971&25.129&4.531&7.551&3.812\\ \hline4.584&0.218&1.751&2.610&22.951&5.006&8.659&5.086\\ \hline3.790&0.264&2.323&3.443&20.571&5.428&9.846&6.780\\ \hline3.133&0.319&3.076&4.516&17.982&5.740&11.084&9.028\\ \hline2.590&0.386&4.060&5.861&15.189&5.865&12.330&11.999\\ \hline2.141&0.467&5.325&7.485&12.215&5.705&13.526&15.903\\ \hline1.770&0.565&6.923&9.331&9.111&5.148&14.608&20.991\\ \hline1.463&0.683&8.883&11.256&5.963&4.075&15.518&27.544\\ \hline1.210&0.827&11.200&13.059&2.882&2.382&16.225&35.865\\ \hline1.000&1.000&13.834&14.560&0.000&0.000&16.730&46.281\\ \hline0.851&1.175&16.259&15.529&-2.253&-2.646&17.023&56.991\\ \hline0.725&1.380&18.819&16.232&-4.264&-5.883&17.220&69.718\\ \hline0.617&1.621&21.473&16.718&-6.040&-9.789&17.348&84.771\\ \hline0.525&1.904&24.191&17.040&-7.589&-14.446&17.429&102.522\\ \hline0.447&2.236&26.951&17.248&-8.928&-19.964&17.478&123.419\\ \hline0.381&2.627&29.739&17.380&-10.079&-26.473&17.507&147.994\\ \hline0.324&3.085&32.543&17.463&-11.065&-34.139&17.521&176.879\\ \hline0.276&3.624&35.358&17.515&-11.908&-43.154&17.526&210.819\\ \hline0.235&4.257&38.180&17.548&-12.627&-53.751&17.548&250.694\\ \hline0.200&5.000&41.006&17.568&-13.241&-66.203&17.568&297.535\\ \hline0.170&5.873&43.834&17.580&-13.763&-80.832&17.580&352.560\\ \hline0.145&6.899&46.664&17.588&-14.208&-98.017&17.588&417.194\\ \hline0.123&8.103&49.495&17.592&-14.587&-118.205&17.592&493.115\\ \hline0.105&9.518&52.327&17.595&-14.910&-141.917&17.595&582.294\\ \hline0.089&11.180&55.159&17.597&-15.185&-169.772&17.597&687.047\\ \hline0.076&13.133&57.991&17.598&-15.419&-202.490&17.598&810.091\\ \hline0.065&15.426&60.823&17.599&-15.618&-240.921&17.599&954.621\\ \hline0.055&18.119&63.656&17.599&-15.788&-286.064&17.599&1124.389\\ \hline0.047&21.283&66.488&17.600&-15.932&-339.089&17.600&1323.801\\ \hline0.040&25.000&69.321&17.600&-16.055&-401.373&17.600&1558.036\\ \hline\end{array}}
Time now (at S=1) or present age in billion years:13.834
'T' in billion years (Gy) and 'D' in billion light years (Gly)

 

[marcus]

One very beautiful thing about this table, as a sample segment of universe history, is that in the distant future one can see the cosmological constant Lambda emerging out of the fog, clearly, as a DISTANCE---a plainly visible concrete thing built into the universe's history.

By convention (going back to before 1920 with Einstein) a small positive Lambda corresponds to a slight negative spacetime curvature---that is a minus one over a large area quantity: the square of a length. So the naturally occurring Lambda constant in the Einstein equation is one over a length squared.

With the usual identification of time and distance, we can simply regard Lambda as a squared growth rate---one over an interval of time, squared. In other words the squared growth rate H_∞² in the equation a couple of posts back is an ALIAS for the cosmological constant Λ in the Einstein equation. (I'm neglecting a stray factor of 3.)

So when you look at the table and see the time quantity 17.6 Gy emerging at around year 60 billion in the future you are seeing a naked manfest appearance of the cosmological constant.

The same as when you see the distance 17.6 billion lightyears emerge, as the distance to the cosmological event horizon, eventually around year 60 billion in the future.
The reciprocal of that distance, squared, is again essentially the cosmological constant (indicating a slight constant negative space-time curvature) that Einstein wrote down in the equation which is now both our law of gravity and our law of geometry.

 

[marcus]

Another beautiful thing the cosmic history calculator shows you is the moment when the recession speed (of any chosen galaxy) stopped slowing down and began to pick up. It is an inflection point on the curve showing the distance to the galaxy. With WMAP numbers (pre-Planck mission 14, 16.5, 3280) this comes around year 7.3 billion.

Let's choose a galaxy which TODAY is at a distance equal to the Hubble radius: 14 billion lightyears.

The table is set to have 26 steps from S=1090 to exact present, and another 26 steps to S=.04.
You can see the minimum recession speed (rightmost column!) comes in the S=1.7 row, around year 7.3 billion.
You can also see that for the sample case we are tracking, where the distance today is 14 Gly, the current Hubble radius, the slowest recession speed ever attained is 0.8516 c. That is about 85% of the speed of light.

At present, because the galaxy is at Hubble radius, the recession speed is exactly c. And as you can also see from the table, in future it will continue to grow.

A galaxy at half the distance (now at 7 Gly instead of 14 Gly) would have a proportionally scaled recession speed history---just divide all the speeds by two! So knowing this one sample history let's us get the recession speeds for objects at other distances as well.

{\begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14&16.5&3280&69.86&0.72&0.28\\ \hline \end{array}} {\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&v_{rec}\\ \hline 1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.001&20.1636\\ \hline 832.918&0.001201&0.000592&0.000983&45.527&0.055&0.074&0.001&17.0999\\ \hline 636.471&0.001571&0.000922&0.001508&45.289&0.071&0.096&0.002&14.5867\\ \hline 486.356&0.002056&0.001428&0.002302&45.009&0.093&0.125&0.003&12.5044\\ \hline 371.647&0.002691&0.002197&0.003500&44.684&0.120&0.163&0.005&10.7630\\ \hline 283.992&0.003521&0.003365&0.005304&44.307&0.156&0.212&0.008&9.2950\\ \hline 217.011&0.004608&0.005131&0.008015&43.872&0.202&0.275&0.013&8.0485\\ \hline 165.828&0.006030&0.007798&0.012088&43.369&0.262&0.357&0.020&6.9840\\ \hline 126.717&0.007892&0.011817&0.018200&42.790&0.338&0.462&0.031&6.0704\\ \hline 96.830&0.010327&0.017862&0.027367&42.124&0.435&0.598&0.047&5.2831\\ \hline 73.992&0.013515&0.026948&0.041109&41.360&0.559&0.773&0.072&4.6026\\ \hline 56.541&0.017686&0.040590&0.061703&40.483&0.716&0.996&0.109&4.0129\\ \hline 43.205&0.023145&0.061058&0.092556&39.477&0.914&1.280&0.167&3.5009\\ \hline 33.015&0.030289&0.091754&0.138771&38.325&1.161&1.640&0.253&3.0557\\ \hline 25.228&0.039638&0.137768&0.207983&37.005&1.467&2.093&0.383&2.6682\\ \hline 19.278&0.051872&0.206718&0.311611&35.494&1.841&2.661&0.580&2.3305\\ \hline 14.731&0.067883&0.310005&0.466715&33.764&2.292&3.365&0.876&2.0363\\ \hline 11.257&0.088835&0.464670&0.698717&31.784&2.824&4.228&1.323&1.7800\\ \hline 8.602&0.116254&0.696135&1.045272&29.520&3.432&5.269&1.994&1.5571\\ \hline 6.573&0.152136&1.042148&1.561411&26.934&4.098&6.502&3.003&1.3641\\ \hline 5.023&0.199093&1.558281&2.325166&23.985&4.775&7.922&4.517&1.1988\\ \hline 3.838&0.260543&2.324459&3.439363&20.641&5.378&9.496&6.782&1.0605\\ \hline 2.933&0.340960&3.450250&5.016065&16.884&5.757&11.146&10.156&0.9516\\ \hline 2.241&0.446198&5.070303&7.113058&12.751&5.689&12.742&15.136&0.8782\\ \hline 1.713&0.583918&7.312958&9.599448&8.373&4.889&14.119&22.363&0.8516\\ \hline 1.309&0.764145&10.232782&12.059647&4.011&3.065&15.144&32.599&0.8871\\ \hline 1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686&1.0000\\ \hline 0.764&1.308652&17.700005&15.230903&-3.469&-4.539&16.147&65.616&1.2029\\ \hline 0.682&1.465878&19.447858&15.566734&-4.731&-6.935&16.236&75.350&1.3183\\ \hline 0.609&1.641994&21.229081&15.819561&-5.879&-9.654&16.301&86.289&1.4531\\ \hline 0.544&1.839269&23.035135&16.007122&-6.919&-12.726&16.348&98.568&1.6086\\ \hline 0.485&2.060245&24.859344&16.144845&-7.857&-16.187&16.380&112.342&1.7865\\ \hline 0.433&2.307770&26.697095&16.244907&-8.700&-20.077&16.402&127.785&1.9889\\ \hline 0.387&2.585034&28.544549&16.317231&-9.457&-24.446&16.416&145.094&2.2179\\ \hline 0.345&2.895609&30.399001&16.369270&-10.135&-29.346&16.425&164.489&2.4765\\ \hline 0.308&3.243498&32.258319&16.406749&-10.742&-34.841&16.430&186.221&2.7677\\ \hline 0.275&3.633183&34.121403&16.433445&-11.285&-41.000&16.433&210.567&3.0952\\ \hline 0.246&4.069687&35.987064&16.452507&-11.770&-47.901&16.453&237.840&3.4630\\ \hline 0.219&4.558633&37.854565&16.466097&-12.204&-55.634&16.466&268.393&3.8759\\ \hline 0.196&5.106324&39.723214&16.475939&-12.592&-64.297&16.476&302.617&4.3390\\ \hline 0.175&5.719816&41.592963&16.482824&-12.938&-74.001&16.483&340.955&4.8582\\ \hline 0.156&6.407015&43.463378&16.487715&-13.247&-84.873&16.488&383.899&5.4403\\ \hline 0.139&7.176777&45.334268&16.491186&-13.523&-97.051&16.491&432.003&6.0926\\ \hline 0.124&8.039020&47.205331&16.493809&-13.769&-110.692&16.494&485.887&6.8235\\ \hline 0.111&9.004857&49.076799&16.495546&-13.989&-125.973&16.496&546.245&7.6425\\ \hline 0.099&10.086732&50.948438&16.496771&-14.186&-143.090&16.497&613.854&8.5601\\ \hline 0.089&11.298588&52.820200&16.497630&-14.361&-162.263&16.498&689.587&9.5881\\ \hline 0.079&12.656041&54.691883&16.498394&-14.518&-183.740&16.498&774.418&10.7395\\ \hline 0.071&14.176583&56.563793&16.498808&-14.658&-207.797&16.499&869.442&12.0295\\ \hline 0.063&15.879808&58.435746&16.499091&-14.783&-234.745&16.499&975.882&13.4745\\ \hline 0.056&17.787665&60.307731&16.499279&-14.894&-264.931&16.499&1095.110&15.0932\\ \hline 0.050&19.924739&62.179573&16.499566&-14.994&-298.742&16.500&1228.663&16.9063\\ \hline 0.045&22.318568&64.051596&16.499641&-15.082&-336.617&16.500&1378.261&18.9374\\ \hline 0.040&25.000000&65.923630&16.499682&-15.162&-379.041&16.500&1545.833&21.2125\\ \hline \end{array}}Time now (at S=1) or present age in billion years: 13.753301
'T' in billion years (Gy) and 'D' in billion light years (Gly), sample recession speed history of matter now at distance R₀, shown as multiples of the speed of light
====

 

[marcus]

Using model parameters from the recent Planck mission report we get nearly the same recession speed history as above.
From Planck, combined with earlier data, we get 14.4 Gly, 17.3 Gly, and 3400. Plugging these parameters into the calculator we get that the minimum recession speed comes at S=1.652 and year 7.592 billion. For a galaxy which is now at current Hubble radius R_o = 14.4 Gly from us, the minimum recession speed is 0.87258c .

So 87% of the speed of light, instead of 85% (as found with earlier model parameters). I think the difference is mainly due to the longer Hubble radius 14.4 instead of 14.0. The representative galaxy we choose to track is slightly more distant, so its recession speeds are slightly higher throughout history, including the minimum.

The minimum is attained somewhat later, namely year 7.6 billion instead of year 7.3 billion which we found in preceding post using 2010 WMAP parameters.

One thing that is easy to do with the table calculator is see what happens when you vary parameters slightly. You can find for instance that the increasing the eventual Hubble radius R_∞ (keeping the other two the same) will make the minimum speed come later.
That makes sense--it delays the onset of "accelerated expansion". A cosmological constant of zero would correspond to infinite Hubble radius, and the expansion speed would continue declining indefinitely and never bottom out. So the longer R_∞ is, the longer you have to wait for acceleration to occur. Accordingly, we see the year of the minimum change from 7.3 to 7.6 billion when we adopt Planck mission numbers and increase R_∞ from 16.5 to 17.3 Gly.

 

[marcus]

There might eventually be a "learner's manual" to go with Jorrie's calculator so I'll experiment with a few cosmic history tables that one can find things into point out and discuss. Here is one that shows the "deja vu" epoch. An earlier time when any galaxy would have the same recession speed that it does right now. This comes around year 3.3 billion. The table also shows a few other points of interest. It runs from S=10 around the time the the first galaxies formed, up to present S=1 and then on to S=0.1 when distances will be ten times what they are now.
I used Planck 2013 model parameters and specified 17 steps from start to present.{\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}'T' in billion years (Gy) and 'D' in billion light years (Gly), a sample recession speed history of matter now at distance R_o is shown in multiples of the speed of light.{\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&v_{rec}sample\\ \hline 10.000&0.100&0.545&0.820&30.684&3.068&4.717&1.558&1.76\\ \hline 8.733&0.115&0.668&1.004&29.536&3.382&5.270&1.916&1.64\\ \hline 7.627&0.131&0.819&1.229&28.307&3.711&5.873&2.354&1.54\\ \hline 6.661&0.150&1.004&1.504&26.994&4.053&6.528&2.893&1.44\\ \hline 5.817&0.172&1.229&1.840&25.591&4.399&7.234&3.554&1.35\\ \hline 5.080&0.197&1.505&2.249&24.093&4.743&7.988&4.364&1.26\\ \hline 4.437&0.225&1.842&2.744&22.495&5.070&8.786&5.357&1.18\\ \hline 3.875&0.258&2.253&3.341&20.794&5.367&9.622&6.573&1.11\\ \hline 3.384&0.296&2.753&4.056&18.988&5.611&10.484&8.061&1.05\\ \hline 2.955&0.338&3.358&4.903&17.077&5.779&11.357&9.877&0.99\\ \hline 2.581&0.387&4.088&5.891&15.065&5.837&12.225&12.089&0.95\\ \hline 2.254&0.444&4.960&7.017&12.963&5.751&13.066&14.775&0.91\\ \hline 1.968&0.508&5.994&8.264&10.788&5.481&13.856&18.023&0.89\\ \hline 1.719&0.582&7.203&9.592&8.567&4.983&14.574&21.929&0.87\\ \hline 1.501&0.666&8.593&10.941&6.334&4.219&15.201&26.597&0.88\\ \hline 1.311&0.763&10.164&12.235&4.132&3.151&15.726&32.134&0.90\\ \hline 1.145&0.873&11.902&13.405&2.002&1.749&16.147&38.655&0.94\\ \hline 1.000&1.000&13.787&14.400&0.000&0.000&16.472&46.279&1.00\\ \hline 0.873&1.145&15.794&15.201&-1.890&-2.164&16.714&55.139&1.08\\ \hline 0.763&1.311&17.896&15.814&-3.607&-4.729&16.888&65.388&1.19\\ \hline 0.666&1.501&20.071&16.267&-5.157&-7.743&17.010&77.200&1.33\\ \hline 0.582&1.719&22.297&16.591&-6.544&-11.249&17.093&90.781&1.49\\ \hline 0.508&1.968&24.561&16.818&-7.775&-15.305&17.149&106.372&1.69\\ \hline 0.444&2.254&26.850&16.974&-8.863&-19.976&17.185&124.252&1.91\\ \hline 0.387&2.581&29.157&17.081&-9.820&-25.343&17.207&144.745&2.18\\ \hline 0.338&2.955&31.475&17.153&-10.660&-31.502&17.220&168.223&2.48\\ \hline 0.296&3.384&33.802&17.202&-11.396&-38.563&17.226&195.115&2.83\\ \hline 0.258&3.875&36.135&17.234&-12.041&-46.654&17.234&225.913&3.24\\ \hline 0.225&4.437&38.470&17.256&-12.605&-55.923&17.256&261.183&3.70\\ \hline 0.197&5.080&40.809&17.271&-13.098&-66.538&17.271&301.571&4.24\\ \hline 0.172&5.817&43.149&17.280&-13.528&-78.695&17.280&347.819&4.85\\ \hline 0.150&6.661&45.490&17.287&-13.905&-92.617&17.287&400.776&5.55\\ \hline 0.131&7.627&47.832&17.291&-14.233&-108.558&17.291&461.415&6.35\\ \hline 0.115&8.733&50.174&17.294&-14.521&-126.813&17.294&530.850&7.27\\ \hline 0.100&10.000&52.516&17.296&-14.772&-147.715&17.296&610.357&8.33\\ \hline \end{array}}
The sample galaxy's present-day recession speed is 1c, the speed of light. Deja vu is at S=3.00,when the galaxy was also receding at the speed of light. The table comes close enough (S=2.955) so that the speed in that row of the table is 0.99c.
Minimum speed occurs around S=1.7. Looking at that row of the table, one can see that for the sample galaxy we've chosen the slowest it ever is, in the whole of cosmic history, is 0.87c, 87% of the speed of light.

In its D_then column the table also shows the radius of the past lightcone. It is the distance of something we are now getting light from at the time it emitted the light. You can see by scanning down the D_then column that the greatest distance at the time of emission is 5.8 billion light years. An emitter at this maximum remove is receding exactly at speed c, so that the light we are receiving from it at first "stood still" (could not close the distance between us) but later began to make headway. D_then coincides with Hubble radius R at that moment in time, as the table also shows.

 

[marcus]

Today I happened to get curious about early times, not the first second or few minutes of the cosmos but something simpler to picture, like year 2000 from the start of expansion. So I put in S=20000.
Ooops, have to go to supper. back later, here's the output for that stretch{\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}} {\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.00005&20000.0&0.00000187&0.00000350&46.177&0.002309&3.21&659.18\\ \hline \end{array}}

You can see it is year 1,870. Just a bit before year 2000. I'll think about what it says conditions were like, after supper.

Part of this is just learning to read off from the table, and get the decimal point in the right place. You know what the temperature of of the CMB is today, around 2.76 kelvin. To get the temperature of radiation back then I guess you just multiply by 20000, or by whatever S is at the time. So 5.5 x 10⁴ kelvin---i.e. around 55,000 kelvin.

And the cube of S is 8 x 10¹². So the density of matter was 8 trillion times what it is today. But that isn't all that much because on average it is so scarce today. amounts to only about 0.23 nanojoule per cubic meter. energy equivalent, including dark matter which is the bulk of it.

So back then, in year 1870, a cubic meter contained 1840 joules worth of matter
1840 joules/c^2 into Google gives: 2 x 10^-11 grams. I can hardly believe it is so little!
Well that is what it seems to be.

 

[Jorrie]

{\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}} {\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.00005&20000.0&0.00000187&0.00000350&46.177&0.002309&3.21&659.18\\ \hline \end{array}}
Part of this is just learning to read off from the table, and get the decimal point in the right place. ...

I find it useful to set the decimals to 9 for such small values, because then the digits represent years or light years. For slightly larger minima, six decimal digits obviously represent My and so on. Sadly, one can't change it halfway through a long table...

 

[Neko]

Long ago in a galaxy far far ...

At what cosmological distance can we be confident the source of the light we see exists today? My son just told me it's spooky action at a distance.

 

[Jorrie]

At what cosmological distance can we be confident the source of the light we see exists today? My son just told me it's spooky action at a distance.

Galaxies are the farthest sources that we are pretty confident that they still exist today, because they are (sort-of) regenerating stars from the gas that that they lock up. The farthest confirmed one that I know of is MACS0647-JD at redshift of 10.9, meaning light took 13.3 billion years to reach us. Due to cosmic expansion, MACS0647-JD must be some 32 billion light years away today.

Potentially farther galaxies are continuously discovered, but it takes some time for the redshifts to be confirmed by other resources.

 

[mfb]

My son just told me it's spooky action at a distance.

This term is used in a completely different context, and has nothing to do with light of old galaxies.

 

[Neko]

Spooky action at a distance

This term is used in a completely different context, and has nothing to do with light of old galaxies.

Jorrie:

Thank you. I understand the term is generally used in a QP context. My son has a sense of humor. He was transcending parsecs and Hubble and red shift. Do you have any thoughts on the existential question?

Neko

 

[mfb]

No philosophy here, please, that usually leads to nothing.

 

[bobie]

It's great to have a Cosmic Event Horizon column!

It is regrettable that Jorrie is inactive at the moment. I hope you can help , marcus.

According to the calculator, the horizon now is just a liitle greater than the radius (14.4 vs 16.4), why so? we know that the ant always reaches its goal, however distant, even if the V_Erubber espansion rate is 100 000 times greater than its own speed v_ant.
Why such a great difference here? Are the formulas different? The conditions seem nearly the same,
nay, much better since recession speed V_EU ≈ V_light.

The link says the formula for D_hor is 1/S ∫^S₀ dS/H,
where can I find the original formula and an explanation?

Thanks

 

[Jorrie]

According to the calculator, the horizon now is just a liitle greater than the radius (14.4 vs 16.4), why so? we know that the ant always reaches its goal, however distant, even if the V_Erubber espansion rate is 100 000 times greater than its own speed v_ant.

The "ant always reaches its goal" only in the case of coasting or decelerating models, because they have an infinite cosmic event horizon radius. In an accelerating model, the Hubble radius (1/H0) always tends towards the event horizon radius as time goes on.

Why such a great difference here? Are the formulas different? The conditions seem nearly the same, nay, much better since recession speed V_EU ≈ V_light.

I do not quite understand your question, but the relationship between the time light took to reach us (observably universe), the Hubble radius and the event horizon is graphically shown in an attachment (graph from calculator)

 

[bobie]

The "ant always reaches its goal" only in the case of coasting or decelerating models, because they have an infinite cosmic event horizon radius. In an accelerating model, the Hubble radius (1/H0) always tends towards the event horizon radius as time goes on.

Hi Jorrie, I am so glad you replied! I have a few questions nobody could answer.
Can you explain the difference between a stretching rubber balloon and the stretching space?
If you refer to the growing rate of expansion, that is really microscopic, even if it did fluctuate it is on the average ≈1/T₀, we can easily consider it stable and V_e ≈ C.
If you refer to other factor please expand on it.

I do not quite understand your question, but the relationship between the time light took to reach us (observably universe), the Hubble radius and the event horizon is graphically shown in an attachment (graph from calculator)

I was referring to the formula of the rubber band,which is so different from the one you are using, but probably when you explain the difference it will all be clarified.

One more thing:
what is and what is the formula for V now/then? In the link I did not find them and nobody could tell me.
For example for S = 1090 V_then is 3.15c does it mean the (apparent) recession speed is 3.15c?
and V_now is 66.18, what does it represent? and what is the formula, 1090 = \sqrt{\frac{c+v}{c-v}}
Thanks a lot, again

 

[Jorrie]

Bobie, the expanding/stretching balloon is no more than a simple analogy to make a small part of cosmology (most importantly, the distance/redshift relationship) easier to understand - space is nothing like rubber! Stick pennies onto a partially inflated balloon and when it is blown up further, the distances between the pennies change according to Hubble's law. That's all there is to it.

If you refer to the growing rate of expansion, that is really microscopic, even if it did fluctuate it is on the average ≈1/T_{0 ...}

No, I think you are confusing the growth of the apparent radius of the observable universe (which grows at c in appropriate units) with the growth of the distance between remote galaxies. The observable universe depends solely on the time since the BB and its radius is not a distance in the true sense of the word. Expansion rates refer to the change of the proper distance between galaxies over time.*

... what is and what is the formula for V now/then? In the link I did not find them and nobody could tell me. For example for S = 1090 V_then is 3.15c does it mean the (apparent) recession speed is 3.15c?

The expansion rate changes drastically over the history of the universe and there is no simple "V now/then" formula, but the V's are readily calculable from the Hubble parameter H against expansion factor and time. Refer to this Wiki that Markus, Mordred and I worked on some time ago.

-J

* See the definition of D_now in the "Show columns definition and selection" (hover over the question mark) in LightCone 7.

 

[Jorrie]

LightCone7zeit has undergone a minor enhancement of the charting function.

The Chart Options area is opened/closed from the main screen. It allows more customizable charts, limited only by what Google Charts will allow.

It only works in the 'zeit version' of LightCone7 at present. If deemed useful, the standard 'billion years version' could also be upgraded with this functionality in future.

--
Regards
Jorrie

 

[Bandersnatch]

Hi, @Jorrie
We've just had a thread in which a poster wanted to see the evolution of Hubble constant with time. @marcus posted a graph from the 7zeit calculator, but those units the calc uses are not the easiest to comprehend for a neophyte. On the other hand, I've noticed that there is no H column available for display in your other calc (light cone 7). There is the reciprocal (Hubble radius) there, so adding Hubble constant should be relatively easy (not that I know anything about programming).
Do you think you could add such a column as an optional selection for further reference in the non-zeit calculator?

 

[Jorrie]

Hi, @Jorrie
We've just had a thread in which a poster wanted to see the evolution of Hubble constant with time. @marcus posted a graph from the 7zeit calculator, but those units the calc uses are not the easiest to comprehend for a neophyte. On the other hand, I've noticed that there is no H column available for display in your other calc (light cone 7). There is the reciprocal (Hubble radius) there, so adding Hubble constant should be relatively easy (not that I know anything about programming).
Do you think you could add such a column as an optional selection for further reference in the non-zeit calculator?

Yes, it is easy and I already have a draft version of the 'standard' LightCone7 with the H column. The reason for not having released it yet is that I have not decided on the units for graphing it - it has rather awkward units; either it is way smaller than 1 (presently 0.069/Gy), or it is way larger than 1 (68 km/s/Mpc). I was thinking about making it H/Ho, which will pitch it around unity for the present epoch, but then it is still pretty small compared to R, T etc.

In tabular form it obviously does not matter too much, but then one of LightCone's greatest features is the charting...

 

[marcus]

My thoughts might not be pertinent, but I'm glad to see you both in this thread. Here's how I might explain to a neophyte.
H(t) is an instantaneous speed-to-size ratio.
It is what you multiply a distance of size D by to get the speed that distance is expanding.

So the clearest way to express H₀ = H(now) is \frac{1}{14.4 Gy}

If you take the distance 14.4 Gly and multiply by that, you get \frac{14.4 Gly}{14.4 Gy} namely the speed of light, which is the right thing.

If you take any other largescale distance and multiply by \frac{1}{14.4 Gy} you get the speed that distance is currently expanding.

 

[marcus]

Maybe it would be possible to get the neophyte to understand that because it is a speed-to-size ratio the natural type of units to express it in is Time^-1.

Once that is understood Jorrie would have a fair amount of freedom in choosing the units compatibly.

 

[Jorrie]

Proposed updated LightCone 7 (standard units) and default column selections. The value H/Ho seems to fit best into the scheme of things...

{\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline T_{Ho} (Gy) & T_{H\infty} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}} {\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{gen}/c&H/Ho \\ \hline 1090.000&0.000373&0.000628&45.331596&0.041589&0.056714&21.023&22915.263\\ \hline 339.773&0.002496&0.003956&44.183524&0.130038&0.178562&10.712&3639.803\\ \hline 105.913&0.015309&0.023478&42.012463&0.396668&0.552333&5.791&613.344\\ \hline 33.015&0.090158&0.136321&38.051665&1.152552&1.651928&3.200&105.633\\ \hline 10.291&0.522342&0.785104&30.917756&3.004225&4.606237&1.782&18.342\\ \hline 3.208&2.977691&4.373615&18.247534&5.688090&10.827382&1.026&3.292\\ \hline 1.000&13.787206&14.399932&0.000000&0.000000&16.472274&1.000&1.000\\ \hline 0.312&32.884943&17.184900&11.117770&35.666086&17.224560&2.688&0.838\\ \hline 0.132&47.725063&17.291127&14.219438&107.785602&17.291127&6.313&0.833\\ \hline 0.056&62.598053&17.299307&15.535514&278.255976&17.299307&14.909&0.832\\ \hline 0.024&77.473722&17.299802&16.092610&681.060881&17.299802&35.227&0.832\\ \hline 0.010&92.349407&17.299900&16.328381&1632.838131&17.299900&83.237&0.832\\ \hline \end{array}}

 

[Jorrie]

Proposed updated LightCone 7 (standard units) and default column selections. The value H/Ho seems to fit best into the scheme of things...

I have uploaded LightCone7s (the 's' for "standard units") if anyone wants to play with it. Not in my sig. yet, since some bug may still lurk somewhere. The main change in the user interface is that the chart options was transferred from LightCone7z and the "set default chart range" has disappeared - it is redundant now. It will also disappear in LightCone7z in due course.
-J

 

[Gaz]

I don't get it.

Note that the CEH is different from the Hubble radius. The Hubble radius is the distance that is growing at rate c. It is currently 13.9 Gly and the CEH (the reachable radius) is 15.6 Gly.
I think you know this but I'll say it just in case others read this.

why is the CEH different from the Hubble?? If the distance between Earth and a galaxy (proper distance) is greater than 13.88 billion light years then the expansion is greater than light can travel so it will never get there ?

 

[Bandersnatch]

I don't get it.
why is the CEH different from the Hubble?? If the distance between Earth and a galaxy (proper distance) is greater than 13.88 billion light years then the expansion is greater than light can travel so it will never get there ?

Consider that the Hubble radius is the reciprocal of the Hubble constant.
Now, what happens to a light signal at the present edge of the Hubble sphere if the Hubble constant goes down with time?

 

[Gaz]

so the Hubble constant is decreasing?

 

[Bandersnatch]

Yes. Look at the table and graph in post #72 above.
The value of interest is $H/H_0$ and how it changes with time - this is the fractional change, so e.g. a value of 2 means twice the present value of $H_0$, which is 67-ish km/s/Mpc.

 

[Gaz]

thanks it makes sense now =)

 

[Mordred]

I have uploaded LightCone7s (the 's' for "standard units") if anyone wants to play with it. Not in my sig. yet, since some bug may still lurk somewhere. The main change in the user interface is that the chart options was transferred from LightCone7z and the "set default chart range" has disappeared - it is redundant now. It will also disappear in LightCone7z in due course.
-J

Thanks Jorrie I'll have to update the link to my webpage when the product is finalized. It's unfortunate I can't use this on another forum I've been actively supporting. (More due to the other sites latex structure). Though I do advertise your product on that forum. I found they needed my help far more than here, as their is plenty of expertise on this forum. I've been of greater help on the other forum.

(If your looking at aspects to add, might I recommend adding density to temperature relations ie the thermodynamic relationships)

 

[Jorrie]

LightCone7s (for "standard"), now appears together with LightCone7z (for "zeit based") in my signature below. Both have the same "LCDM-engine", the only difference being the units being worked in. Billion years (Gy) for "version s" and zeit for "version z". One zeit is simply time divided by 17.3 Gy, a natural timescale of the LCDM model.

The update from '7' to '7s' is about flexibility in specifying the graph formats, as can be seen under the button "Open Cart Options" (the green area of the LightCone7 (partial) screen shot shown below).

 

[Jorrie]

(If your looking at aspects to add, might I recommend adding density to temperature relations ie the thermodynamic relationships)

Eventually, there is a beta-test version available with some additions on density, density parameters and temperature. It is not the 'Forum official' version yet, but it has other interesting changes. E.g. inputs or now more standard - I have done away with Hubble times as input parameters, because they are not the ones used in the literature. Prime inputs are now the Hubble constant in conventional units, the total density parameter and the radiation-matter equality redshift parameters. The matter density parameter is then still a derived value.

The range of calculations are also requested as the more conventional redshift (z) in lieu of the simpler, but less well known "S" parameter. Lastly, the output scaling option for "Zeit" has been left out, since it is a potentially non-standard distraction. I hope the updates will enhance the use of the calculator in the educational field.

The latest beta-test version is available as: LightCone7-2017-01-26.

Edit: we found an error in Omega-calculations of this version. See the thread https://www.physicsforums.com/threads/evolution-of-the-energy-density-parameters.901681/
The corrected version is: LightCone 7, Cosmo-Calculator (2017-1).

Comments/suggestions welcome. I will start a new thread to discuss some of the more subtle aspects of the density parameter calculations.

 

[Mordred]

The usage parameters such as ones found in intro level textbooks is probably the most familiar approach and the one that will probably gain the most usage. The parameters you mentioned being the key ones. People are more familiar with redshift than stretch for example. I agree the best approach should be literature based.

I should have time to help update the user manuals when the testing is done if you'd like my help again on that. I still remember how to edit and text on wikidot

 

[Jorrie]

After some more comments and further testing, it seems like the updated calculator has stabilized on this version: LightCone7-2017-01-30.
I suggest that we leave it for another week in 'testing mode' and then I will 'release' it into the same url as the previous release, so that no links/sig's need to be updated.

 

[Mordred]

I've been running various tests as time allows. I haven't found any issues that I can see thus far

 

[Jorrie]

I've been running various tests as time allows. I haven't found any issues that I can see thus far

Thanks for your effort, Mordred. I have used a specific set of columns as default to highlight the new features, but it may now be time to choose a more general set. It should still be limited so as to not being frightening to newcomers.

Any suggestions.

 

[Jorrie]

I have now changed the link in my Sig below to the latest version that we have tested, with a very basic set of columns as the default, i.e.

{\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&T (Gy)&R (Gly)&D_{now} (Gly)&Temp(K) \\ \hline 1.09e+3&3.72e-4&6.27e-4&4.53e+1&2.97e+3\\ \hline 3.39e+2&2.49e-3&3.95e-3&4.42e+1&9.27e+2\\ \hline 1.05e+2&1.53e-2&2.34e-2&4.20e+1&2.89e+2\\ \hline 3.20e+1&9.01e-2&1.36e-1&3.81e+1&9.00e+1\\ \hline 9.29e+0&5.22e-1&7.84e-1&3.09e+1&2.81e+1\\ \hline 2.21e+0&2.98e+0&4.37e+0&1.83e+1&8.74e+0\\ \hline 0.00e+0&1.38e+1&1.44e+1&0.00e+0&2.73e+0\\ \hline -6.88e-1&3.30e+1&1.73e+1&1.12e+1&8.49e-1\\ \hline -8.68e-1&4.79e+1&1.74e+1&1.43e+1&3.59e-1\\ \hline -9.44e-1&6.28e+1&1.74e+1&1.56e+1&1.52e-1\\ \hline -9.76e-1&7.77e+1&1.74e+1&1.61e+1&6.44e-2\\ \hline -9.90e-1&9.27e+1&1.74e+1&1.64e+1&2.73e-2\\ \hline \end{array}}

There are now a total of 18 selectable columns, including the actual density against redshift and also the various density parameters (the Omegas).
It is very easy to change the default columns in the program, so please let me know if you want to see other columns as default.
The idea of a small selection is to not overwhelm newcomers with too much data.

 

[Jenab2]

The new A20 tabular calculator let's you look at changing geometry out to about 88 billion years according to the standard LCDM cosmic model (with usual estimates for the parameters.).
http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A20.html

It's pretty neat. Here is one sample tabulation. Red stuff is just the three standard parameters, estimated based on observation. No reason to change them, although in this calculator you CAN change them and play around to see the effects.
The blue stuff is what I put into give bounds and step size for the table I wanted it to generate
From the present (S=1) to the distant future (S=0.01) when distances are 100 times what they are today. In steps of ΔS = 0.09. those are just what I chose. If you choose a smaller step size like ΔS = 0.01 you get a table with more rows, like around 100 rows instead of only 12 rows. I won't bother to align the columns. It's probably legible as is.
===quote===

Hubble time now (Ynow) 13.9 Gy Change as desired (9 to 16 Gy)
Hubble time at infinity (Yinf) 16.3 Gy Change as desired (larger than Ynow)
Radiation and matter crossover (S_eq) 3350 Radiation influence (inverse: larger means less influence)
Upper limit of Stretch range (S_upper) 1.0 S value at the top row of the table (equal or larger than 1)
Lower limit of Stretch range (S_lower) 0.01 S value at the bottom row of table (S_lower smaller than S_upper)
Step size (S_step) 0.09 Step size for output display (equal or larger than 0.01)

Stretch (S) Scale (a) Time (Gy) T_Hubble (Gy) D_now (Gly) D_then (Gly)
1.000 1.000 13.769 13.896 0.000 0.000
0.910 1.099 15.104 14.387 -1.219 -1.339
0.820 1.220 16.630 14.829 -2.536 -3.093
0.730 1.370 18.374 15.221 -3.884 -5.320
0.640 1.563 20.402 15.545 -5.270 -8.234
0.550 1.818 22.772 15.812 -6.676 -12.138
0.460 2.174 25.618 16.006 -8.108 -17.627
0.370 2.703 29.120 16.143 -9.555 -25.825
0.280 3.571 33.629 16.233 -11.010 -39.323
0.190 5.263 39.934 16.278 -12.474 -65.650
0.100 10.000 50.390 16.296 -13.939 -139.393
0.010 100.000 87.919 16.300 -15.406 -1540.607

For the model used, see this thread on Physicsforums.
=====endqquote=====

what this tells you, among other things, is which of the galaxies out there you can reach if you flash a signal to them today.

It says ANYTHING THAT IS TODAY NEARER THAN 15.4 BILLION LY is a target you can reach if you flash a message today, and it will get there WITHIN 88 BILLION YEARS.

It also says that 88 billion years from now is when distances will be 100 times what they are today (cosmological distances, not dimensions of bound structures like a rock or solar system)

So if you select a galaxy which is today 15.4 billion LY and you flash a message today, when the message finally gets there the distance to the galaxy (and the message arriving at it) will be 1540 billion LY.
You can read that off the table too.

Is there anyone to whom this does NOT make sense. This is a great calculator and an interactive version of the standard cosmic model that is in professional use (LCDM) and there must be plenty of people who can explain if you find anything obscure about the table. Everybody should get so they understand the table outputs of this calculator both of past history and of the future, IMHO. They are basic.

This is the actual distance of the object (15.4 billion light years), not the distance we would observe it at, which would be nearer to 8 billion light years. The 15.4 Gly has already been corrected for the expansion of the universe since the light we currently can see was emitted.